Hi, everyone! Is there any efficient way to simplify the following tensor product

$X \otimes X + X^T \otimes X^T$, where $X$ is a square $n \times n$ matrix.

My goal is to efficiently compute the dominant eigenvector of $X \otimes X + X^T \otimes X^T$. However, the direct way is computationally expensive. Is it possible to simplify it avoid tensor computation.

For example, to compute the dominant eigenvector of $X \otimes X$, i can compute the dominant eigenvector of $X$ first, denoted by $v$. Then i only need to compute $v \cdot v^T$, which is the dominant eigenvector of $X \otimes X$. However, when it comes to $X \otimes X + X^T \otimes X^T$, i have no idea. Could anyone give me some suggestion or reference? Thank you in advance!